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Law of Constant Interfacial Angles Angles Between the Crystal Faces of a Given Species Are Constant  Contact (A

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Law of Constant Interfacial Angles Angles Between the Crystal Faces of a Given Species Are Constant  Contact (A CSIR-National Chemical Laboratory, Pune – 411008 Crystallography deals with crystals A crystal is a solid with an orderly, repeating arrangement of atoms 2 Crystals are everywhere! 3 Hexagonal System, Sp. Gp. P63/mmc Crystals are everywhere! Transparent Opaque Translucent 4 Crystals are everywhere! 5 fibrous Needles Prismatic Tabular 6 Crystals found in everyday life 7 Crystals found in everyday life Diamond Graphite Fullerene 8 Historic definition before the advent of crystallography -A solid with well-defined faces Crystallographic definition - A material with a regularly repeating structural motif The strict definition is more vague - Any material that gives diffraction pattern with sharp peaks 9 What is a Crystal? 10 What is not a Crystal? 11 What is a Crystal? A Regular Arrangements of Atoms, Ions or Molecules 12 CLASSIFICATION OF SOLIDS BASED ON ATOMIC ARRANGEMENT AMORPHOUS QUASICRYSTALS CRYSTALS Short range order order long range order + + + No periodicity No periodicity periodicity What? Science of the arrangement of atoms in molecules Science of the arrangement of molecules with respect to each other Implications for these arrangements for a myriad of areas of science from bioscience to nanotechnology Not the science of crystals 14 Why? Materials’ properties are intimately related to their structures Understanding certain properties requires knowledge of atomic arrangement, e.g. piezoelectric, polarization, optical activity, hardness, compressibility, solubility, colour , density etc. This technique is essential for chemist’s today for knowledge of accurate molecular structure, which is an essential for structure based functional studies to aid in the development of effective materials. 15 Geometrical Crystallography- Study of external shape of crystal 16 Early thoughts about crystals Symmetry Crystals are solid - but solids are not necessarily crystalline Crystals have symmetry (Kepler,1611) and long range order - First speculation on the nature of six-fold symmetry of snowflakes based on the observation that small ice spheres can produce a regular polygon 17 Group discussion Kepler wondered why snowflakes have 6 corners, never 5 or 7. By considering the packing of polygons in 2 dimensions, demonstrate why pentagons and heptagons shouldn’t occur. Empty space is not allowed 18 ? Empty space is not allowed ? Empty space is not allowed Space filling repeat patterns Only 2, 3, 4 and 6-fold rotations can produce space filling patterns Crystal faces: crystal usually Crystal edge: It is formed by bounded by a number of flat intersection of two adjacent surfaces (faces). faces Face 4 + 4 + 4 = 12 edge External parts of crystal Solid angle: It is formed by Interfacial Angle: It is angle between two intersection of more than two adjacent faces. adjacent faces More accurately it is angle locates between two verticals drawn to any two adjacent faces. Interfacial Angle 4 + 4 = 8 Solid angle Interfacial Angle Law of constant interfacial angles Angles between the crystal faces of a given species are constant Contact (A. Carangeot,1783): To determine the angle between two surfaces, one has to hold the crystal edge at the scissor opening between the limbs of the goniometer. The angle being measured is read from the scale. Reflecting (W.H. Wollaston, 1809): Instead of measuring the angle formed by the meeting of two faces of a crystal directly, it measured the angle formed by the meeting of rays of light reflected from them. 29 CRYSTAL SYSTEMS are divided into 7 main groups. The first group is the ISOMETRIC (Cubic). This literally means “equal measure” and refers to the equal size of the crystal axes. Fluorite Crystals Pyrite Crystals Spinel Fluorite Pyrite Cube with Pyritohedron Octahedron Cube Striations Garnet Trapezohedron Garnet - Dodecahedron Grossularite, Dodecahedra Three horizontal axes meeting at angles of 120o and one perpendicular axis. This model represents a hexagonal PRISM (the outside hexagon - six sided shape). The top and bottom faces are called PINACOIDS and are perpendicular to the vertical “c” axis. BERYL 34 WULFENITE APOPHYLLITE on Stilbite Topaz from Topaz Mountain, Utah. Pinacoid View STAUROLITE Prism View Top View Mica Orthoclase Gypsum Microcline, variety Amazonite Abbé René Just Haűy (1743 – 1822): Father of Crystallography Gabriel Delafosse Auguste Laurent Christian Samuel Weiss Crystal Structure Model Symmetry in Crystal Bravais, a graduate of the École Polytechnique and a professor of physics, worked out a mathematical theory of crystal symmetry based on the concept of the crystal lattice, of which there were 14. The Unit Cell “The smallest repeat unit of a crystal structure, in 3D, which shows the full symmetry of the structure” The unit cell is a box with: • 3 sides - a, b, c • 3 angles - , , Only 1/8 of each lattice point in a unit cell can actually be assigned to that cell. Each unit cell in the figure can be associated with 8 x 1/8 = 1 lattice point. 42 Building a crystal vc 2c ub c 2b b 0 a 2a 3a 4a ta 43 Lattice Crystal Translationally periodic Translationally periodic arrangement of points in arrangement of motifs space Crystal = Lattice + Motif Lattice the underlying periodicity of the crystal Basis or Motif atom or group of atoms associated with each lattice points Lattice how to repeat Motif what to repeat Crystal structure can be obtained by attaching atoms, groups of atoms or molecules which are called basis (motif) to the lattice sides of the lattice point. + = Elemental solids (Argon): Basis = single atom. Polyatomic Elements: Basis = two or four atoms. Complex organic compounds: Basis = thousands of atoms. Seven unit cell shapes Triclinic abc 90° Monoclinic abc ==90°, 90° Orthorhombic abc ===90° Tetragonal a=bc ===90° Rhombohedral a=b=c ==90° Hexagonal a=bc ==90°, =120° Cubic a=b=c ===90° The 14 Bravais lattices Triclinic P Monoclinic P Monoclinic ( C ) a b c, a b c, 90 90 a b c, 90oo , ß 90 The 14 Bravais lattices Orthorhombic (P) Orthorhombic (C) Orthorhombic (I) Orthorhombic (F) o a b c, =ß= =90 a b c, =ß= =90o a b c, =ß= =90o a b c, =ß= =90o The 14 Bravais lattices Rhombohedral (R) Trigonal P a=b=c, =ß= 90o Tetragonal (P) Tetragonal (I) Hexagonal (P) oo a=b c, =ß= =90o a=b c, =ß= =90o a=b c, =ß=90 , =120 The 14 Bravais lattices Cubic (P) Cubic (I) Cubic (F) o a=b=c, =ß= =90o a=b=c, =ß= =90o a=b=c, =ß= =90 “Symmetry elements define the (conceptual) motion of an object in space, the symmetry operation, leads to an arrangement that is indistinguishable from the initial arrangement.‖ Rotation, reflection and inversion operations generate a variety of unique arrangements of lattice points (i.e., a shape structure) in three dimensions. Rotational symmetry Rotation about an axis: 1, 2, 3, 4 or 6 56 Rotational symmetry ESCHER’S DRAWING 57 Mirror Plane Symmetry “Arises when one half of an object is the mirror image of the other half” m 58 Mirror Symmetry Right and left hands are identical by reflection through a mirror plane. Imagine σ as a plane pointing into the page. This mirror plane is the symmetry element. The motion of taking one hand through the plane to give its reflection is the symmetry operation. Centre of Symmetry (i) “present if you can draw a straight line from any point, through the centre, to an equal distance the other side, and arrive at an identical point” Centre of symmetry at S 60 The inversion, i Centre of symmetry ◦ reflection through the centre of the molecule to an equal distance on the opposite site. Point of inversion, i The combination of all available symmetry operations (32 point groups), together with translation symmetry, within the all available lattices (14 Bravais lattices) lead to 230 Space Groups that describe the only ways in Arthur Moritz Schönflies (1853-1928) which identical objects can be arranged in an infinite lattice. The International Tables list those by symbol and number, together with symmetry operators, origins, reflection conditions, and space group projection diagrams. Yevgraf Stepanovich Federov (1853-1919) Reflection Plane: Glides The glide plane is perpendicular to the page. A glide consists of a reflection followed by a translation. Rotation Axis: Screw William Hyde Wollaston. Barlow’s theories of the properties of crystals were based on the close packing of atoms.‖ Independently of Schönflies and Federov , Barlow derived the 230 space groups. William T. Hosler, ―Barlow, William,‖ Complete Dictionary of Scientific Biography. 2008. Encyclopedia.com. 20 May, 2012. http://www.encyclopedia.com Nov., 1895: W. Röntgen discovered that when certain substances are exposed to the beam of a cathode ray tube, a new kind of penetrating ray capable of fogging photographic plates even when shielded was emitted – He called it "x-rays". These x-rays also ionized gases through which they passed. Bertha Röntgen’s Hand 8 Nov, 1895 Röntgen, director of the physics laboratory. Arnold Sommerfeld, Director of the Institute for Theoretical Physics. Experimental work on wave-nature (and wave length) of x-rays. Paul von Groth, professor of mineralogy, world renowned authority on crystallography and mineralogy. Interested in atomic/molecular meaning of crystal structure. Paul Peter Ewald, student of Sommerfeld, working on propagation of x-rays in single crystals. Max von Laue, Provatdozent in Sommerfeld’s Institute. The first kind of scatter process to be recognised was discovered by Max von Laue who was awarded the Nobel prize for physics in
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